# Domain

$PlonK$ needs a domain to encode the circuit on (for each point we can encode a row/constraint/gate). We choose a domain $H$ such that it is a multiplicative subgroup of the scalar field of our curve.

## 2-adicity

Furthermore, as FFT (used for interpolation, see the section on FFTs) works best in domains of size $2_{k}$ for some $k$, we choose $H$ to be a subgroup of order $2_{k}$.

Since the pasta curves both have scalar fields of prime orders $p$ and $q$, their multiplicative subgroup is of order $p−1$ and $q−1$ respectively (without the zero element). The pasta curves were generated specifically so that both $p−1$ and $q−1$ are multiples of $2_{32}$.

We say that they have *2-adicity* of 32.

Looking at the definition of one of the pasta fields in Rust you can see that it is defined specifically for a trait related to FFTs:

`#![allow(unused)] fn main() { impl FftParameters for FqParameters { type BigInt = BigInteger; const TWO_ADICITY: u32 = 32; #[rustfmt::skip] const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([ 0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056 ]); }`

The 2-adicity of 32 means that there’s a multiplicative subgroup of size $2_{32}$ that exists in the field.
The code above also defines a generator $g$ for it, such that $g_{2_{32}}=1$ and $g_{i}=1$ for $i∈[[1,2_{32}−1]]$ (so it’s a **primitive** $2_{32}$-th root of unity).

[Lagrange’s theorem](https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)) tells us that if we have a group of order $n$, then we’ll have subgroups with orders dividing $n$. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2.

To find any of these groups, it is pretty straight forward as well. Notice that:

- let $h=g_{2}$, then $h_{2_{31}}=g_{2_{32}}=1$ and so $h$ generates a subgroup of order 31
- let $t=g_{2_{2}}$, then $t_{2_{30}}=g_{2_{32}}=1$ and so $t$ generates a subgroup of order 30
- and so on…

In arkworks you can see how this is implemented:

`#![allow(unused)] fn main() { let size = n.next_power_of_two() as u64; let log_size_of_group = ark_std::log2(usize::try_from(size).expect("too large")); omega = Self::TWO_ADIC_ROOT_OF_UNITY; for _ in log_size_of_group..Self::TWO_ADICITY { omega.square_in_place(); } }`

this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book.

## Efficient computation of the vanishing polynomial

For each curve, we generate a domain $H$ as the set $H=1,ω,ω_{2},⋯,ω_{n−1}$.
As we work in a multiplicative subgroup, the polynomial that vanishes in all of these points can be written and efficiently calculated as $Z_{H}(x)=x_{∣H∣}−1$.

This is because, by definition, $ω_{∣H∣}=1$. If $x∈H$, then $x=ω_{i}$ for some $i$, and we have:

$x_{∣H∣}=(ω_{i})_{∣H∣}=(ω_{∣H∣})_{i}=1_{i}=1$

This optimization is important, as without it calculating the vanishing polynomial would take a number of operations linear in $∣H∣$ (which is not doable in a verifier circuit, for recursion). With the optimization, it takes us a logarithmic number of operation (using exponentiation by squaring) to calculate the vanishing polynomial.